Using the notation introduced in Section 2.1, we now consider the general zero range process on TdN given by generator (2.2). Let us make the following assumptions on the rate functiong :N→[0,∞).
Assumption 1. (i) Non-degeneracy: Assume that g satisfies g(0) = 0 and g(n)>0 for all n >0.
(ii) Lipschitz-property: We require that g is Lipschitz continuous with 0≤ |g(n+ 1)−g(n)| ≤g∗ <+∞
for all n∈N.
(iii) Spectral gap: We also assume that there exist n0 >0 and δ >0 such that g(n)−g(j)≥δ
for any j ∈N and n≥j+n0.
(iv) Attractivity: Let the jump rateg be monotonously increasing, i.e.
g(n+ 1)≥g(n) for all n∈N.
Remark 2.2.1. Let us comment on the different parts of Assumption 1: Part (i) is essential to avoid degeneracies of the particle system. We need the spectral gap property
(iii) in order to prove an explicit uniform in time rate of convergence, since it allows us to quantify the local relaxation to equilibrium of the particle system as well as the global convergence to equilibrium on the level of the limit equation. It implies in particular that g(n)≥g0n with g0 >0. The attractivity (iv) is important to obtain moment–bounds on the particle system, see Subsection 2.6.1. The question of moment bounds is still an open problem in its absence. Assumption (ii) is used at several points in the proof, but could possibly be replaced using the uniform moment bounds originating from assumption (iv) - however, it would affect our strategy to prove the regularity result in several dimensions, see Section 2.7.
In the context of the zero range process with diffusive scaling, two very well-known meth- ods of proving a hydrodynamic limit are the entropy method due to Guo, Papanicolaou, and Varadhan [36], see Theorem 2.2.2, and the relative entropy method due to Yau [86], see Theorem 2.2.3. For an extensive account of these methods in the context of zero range process, see [47].
In order to proceed, we need one more definition. Let µ, ν ∈ P(XN) be two probability measures. Then the relative entropy of µ relative toν is defined as
(2.9) HN(µ|ν) =
Z
XN
log dµdν dµ
whenever µis absolutely continuous with respect to ν. The relative entropy is connected to the Fisher information
(2.10) DN(µ|ν) =
Z
XN
qdµ dνGN
qdµ dνdν.
The entropy method can be summarized in the following theorem. Note that we have not taken great care to optimize the assumptions. The proofs under the assumptions given below can be found in [47].
Theorem 2.2.2 (Guo, Papanicolaou, Varadhan). Assume (i) and (ii) of Assumption 1 as well as g(n)≥g0n for some g0 >0 and let µN0 ∈P(XN) and f0 ∈L∞(Td) such that
N→∞lim PµN0 |hαNη , ϕi − hf0, ϕi|>
= 0,
for every continuous function ϕ∈ C(Td) and every > 0. Furthermore we assume that the initial data satisfy the bounds
HN(µN0 |νρN)≤CNd and
µN0 , 1 Nd
X
x∈TdN
η(x)2
≤C
for some ρ >0 and a constant C <+∞.
Then, for every t ≥ 0, every continuous function ϕ ∈ C(Td), and every > 0, it holds that
N→∞lim PµNt |hαNη , ϕi − hft, ϕi|>
= 0,
where ft is the unique weak solution to (2.3) and µNt solves (2.1) with Cauchy datum µN0 . Thus the entropy method yields propagation of the hydrodynamic profile. The relative entropy method by Yau, on the other hand, concerns the conservation of a stronger notion.
In analogy to (2.6), we define a local Gibbs measure with macroscopic profile ft ∈C(Td) by
(2.11) νfN
t(·)(η) = Y
x∈TdN
σ(ft(Nx))η(x) g(η(x))!Z(σ(ft(Nx))).
This measure has the property that it is locally (in infinitesimal macroscopic neighbour- hoods whereft is constant) in equilibrium with a macroscopic non-equilibrium profile ft asN → ∞. The relative entropy method then yields the following theorem.
Theorem 2.2.3(Yau). Assume (i) and (ii) of Assumption 1 as well as that the partition function Z(·) is finite on all [0,∞), e.g. g(n) ≥ g0n for some g0 > 0. Furthermore, assume that the solution ft to (2.3)satisfies ft ∈C2(Td)and letµNt ∈P(XN) solve (2.1).
Finally assume that initially at t = 0, the relative entropy HN(µN0 |νfN
0(·)) vanishes in the limit, i.e.
lim
N→∞
1
NdHN µN0 |νfN
0(·)
= 0.
Then it holds that
(2.12) lim
N→∞
1
NdHN µNt |νfN
t(·)
= 0 for every t≥0.
Note that the convergence of the relative entropy (2.12) implies that µNt has profile ft, i.e.
N→∞lim PµNt |hαNη , ϕi − hϕ, fti|>
= 0.
Thus the convergence of the relative entropy can be thought of as a stronger notion of the hydrodynamic limit. Yau’s relative entropy method shows that this stronger notion is conserved by the evolution.
Remark 2.2.4. It appears that using a quantitative replacement lemma, see Section 2.6, this result can be translated to a quantitative result of the form
HN µNt |νfNt(·)
≤Ceγ−1tHN µN0 |νfN0(·)
+tr(N),
where limN→∞r(N) = 0 if γ is sufficiently small, and r(N) can be made explicit (al- though, to our knowledge, such a result has never been published). Thus it seems that a quantitative estimate on the rate of convergence is available in the stronger form of the hydrodynamic limit given by the convergence of the entropy relative to the local Gibbs state. However, this convergence is not uniform in time, since γ might be very small.
Therefore even if one manages to prove exponential decay in time of the relative entropy HN µNt |νfN
t(·)
≤ Ce−λt, e.g. by employing a logarithmic Sobolev inequality, it is still not possible to conclude uniform in time convergence if λ < γ−1. In the context of the zero range process, the following logarithmic Sobolev inequality holds [27, 59]:
(2.13) HN(µ|νN,K)≤CN2DN(µ|νN,K)
uniformly in N, K, and µ ∈ P(XN), where we recall that νN,K denotes the canoni- cal measure (2.8). Logarithmic Sobolev inequalities are very effective tools to describe concentration of measure and have been employed widely starting with the works [5, 33].
For a related model, the Ginzburg-Landau model with Kawasaki dynamics, there exists an additional method due to Grunewald, Otto, Villani, and Westdickenberg [34], who prove a logarithmic Sobolev inequality and hydrodynamic limit based on a coarse-graining of the state-space. In principle, it should be possible to extend their method to obtain a uniform rate of convergence. On the other hand, it is not clear how to extend the method to the zero range process and how to obtain uniform-in-time convergence.